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12 QUOTIENT GROUPS 55
all p J> n and q *> n. Since J is right-invariant, this means that for all y e xp
the intersection of xq and the neighborhood Vn j; is not empty. We can then
define, by induction on «, a sequence (xn) in G such that xn e xn and
xn+1 G Vn^cn for all «. Hence it follows, by induction on/?, that for allp > 0 we
have xn+peVn+p_lVn+p_2-'Vn+1VnxnciVn^xn (because V*+lcVn for
all n). Hence the sequence (xn) is a right Cauchy sequence in G. Q.E.D.
Let G be a topological group, and let E be a topological space on which G
acts continuously and transitively. Let x e E and let S^ be the stabilizer of x in
G; then (12.10.1) the continuous mapping hx : si>s x of G into E, which by
hypothesis is surjective, factorizes canonically as
(12.11.4) hx: G 2+ G/$x % E,
where nx is the canonical mapping of G onto the homogeneous space G/SX,
and/x is the bijection s$x\-+s x.
It follows from (12.10.6) that fx is a continuous bijection, but it is not
necessarily a homeomorphism (Section 12.12, Problem 2).
(12.11.5) With the notation above, the bijection fx : G/SX -> E is a homeomor-
phism for each x E E if and only if there exists a point xQeE such that the
mapping hXQ: s \-> s XQ transforms each neighborhood of e in G into a neigh-
borhood of x0 in E.
In view of (12.10.6) we need only prove that the condition is sufficient.
Notice first that each xeE can by hypothesis be written in the form
x = t - x0 for some t e G. If V is a neighborhood of e, it follows from the
hypothesis that V - x = (Vf) x0 is a neighborhood of x, because we can
write (Vf) x0 = f ((f ~JVf) JCQ), and t ~lVt is a neighborhood of e in G
(12.8.3), and y*-+s-y is a homeomorphism of E onto E, for each seG
(12.10.3). Since each open set in G/SX is the image under nx of some open set
in G, it is enough to show that for each open set U in G, and each x e E, the
set hx(U) =fx(nx(U)) = U x is open in E. Now, for each fell, the set
t ~l\J is a neighborhood of e, and therefore (t ~1U) - x is a neighborhood of x
in E, from the first part of the proof. Hence U x = t - ((t -1U) x) is a
neighborhood of t x; by (3.6.4), the proof is complete.
12. QUOTIENT GROUPS
If H is a normal subgroup of a group G, then xH = Ex for each x e G and
therefore the sets G/H and H\G are identical; and G/H has a group structure2.9.1 and 12.8.3). Let (BB) be a